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2.11 THE MOLECULAR NATURE OF INTERNAL ENERGY
Internal energy is energy at the molecular level. The molecular description of internal energy is outside the scope of thermodynamics, but a qualitative understanding of molecular energies is helpful.
Consider first a gas. The molecules are moving through space. A molecule has a translational kinetic energy mv2, where m and v are the mass and speed of the mole-cule. A translation is a motion in which every point of the body moves the same dis-tance in the same direction. We shall later use statistical mechanics to show that the total molecular translational kinetic energy Utr,mof one mole of a gas is directly pro-portional to the absolute temperature and is given by [Eq. (14.14)] Utr,m RT, where R is the gas constant.
If each gas molecule has more than one atom, then the molecules undergo rota-tional and vibrarota-tional motions in addition to translation. A rotation is a motion in which the spatial orientation of the body changes, but the distances between all points in the body remain fixed and the center of mass of the body does not move (so that there is no translational motion). In Chapter 21, we shall use statistical mechanics to show that except at very low temperatures the energy of molecular rotation Urot,m in one mole of gas is RT for linear molecules and RT for nonlinear molecules [Eq. (21.112)]: Urot,lin,m RT; Urot,nonlin,m RT.
Besides translational and rotational energies, the atoms in a molecule have vibra-tional energy. In a molecular vibration, the atoms oscillate about their equilibrium po-sitions in the molecule. A molecule has various characteristic ways of vibrating, each way being called a vibrational normal mode (see, for example, Figs. 20.26 and 20.27).
Quantum mechanics shows that the lowest possible vibrational energy is not zero but is equal to a certain quantity called the molecular zero-point vibrational energy (so-called because it is present even at absolute zero temperature). The vibrational energy contribution Uvib to the internal energy of a gas is a complicated function of temperature [Eq. (21.113)]. For most light diatomic (two-atom) molecules (for ex-ample, H2, N2, HF, CO) at low and moderate temperatures (up to several hundred kelvins), the average molecular vibrational energy remains nearly fixed at the zero-point energy as the temperature increases. For polyatomic molecules (especially those with five or more atoms) and for heavy diatomic molecules (for example, I2) at room temperature, the molecules usually have significant amounts of vibrational energy above the zero-point energy.
Figure 2.14 shows translational, rotational, and vibrational motions in CO2. In classical mechanics, energy has a continuous range of possible values. Quantum mechanics (Chapter 17) shows that the possible energies of a molecule are restricted to certain values called the energy levels. For example, the possible rotational-energy val-ues of a diatomic molecule are J(J 1)b [Eq. (17.81)], where b is a constant for a given molecule and J can have the values 0, 1, 2, etc. One finds (Sec. 21.5) that there is a dis-tribution of molecules over the possible energy levels. For example, for CO gas at 298 K, 0.93% of the molecules are in the J 0 level, 2.7% are in the J 1 level, 4.4% are in the J 2 level, . . . , 3.1% are in the J 15 level, . . . . As the temperature increases, more molecules are found in higher energy levels, the average molecular energy increases, and the thermodynamic internal energy and enthalpy increase (Fig. 5.11).
Besides translational, rotational, and vibrational energies, a molecule possesses electronic energy eel (epsilon el). We define this energy as eel⬅ eeq eq, where eeqis the energy of the molecule with the nuclei at rest (no translation, rotation, or vibration) at positions corresponding to the equilibrium molecular geometry, and eq
is the energy when all the nuclei and electrons are at rest at positions infinitely far apart from one another, so as to make the electrical interactions between all the charged particles vanish. (The quantity eqis given by the special theory of relativity
3 2
3 2
3 2 1
2
Section 2.11 The Molecular Nature
of Internal Energy 67
O C O
A translation
O C O
A rotation
O C O
A vibration
Figure 2.14
Kinds of motions in the CO2 molecule.
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The First Law of Thermodynamics 68
as the sum of the rest-mass energies mrestc2 for the electrons and nuclei.) For a sta-ble molecule, eeqis less than eq.
The electronic energy eelcan be changed by exciting a molecule to a higher elec-tronic energy level. Nearly all common molecules have a very large gap between the lowest electronic energy level and higher electronic levels, so at temperatures below, say, 5000 K, virtually all the molecules are in the lowest electronic level and the con-tribution of electronic energy to the internal energy remains constant as the tempera-ture increases (provided no chemical reactions occur).
In a chemical reaction, the electronic energies of the product molecules differ from those of the reactant molecules, and a chemical reaction changes the thermody-namic internal energy U primarily by changing the electronic energy. Although the other kinds of molecular energy generally also change in a reaction, the electronic energy undergoes the greatest change.
Besides translational, rotational, vibrational, and electronic energies, the gas molecules possess energy due to attractions and repulsions between them (intermo-lecular forces); intermo(intermo-lecular attractions cause gases to liquefy. The nature of inter-molecular forces will be discussed in Sec. 21.10. Here, we shall just quote some key results for forces between neutral molecules.
The force between two molecules depends on the orientation of one molecule rel-ative to the other. For simplicity, one often ignores this orientation effect and uses a force averaged over different orientations so that it is a function solely of the distance r between the centers of the interacting molecules. Figure 21.21a shows the typical be-havior of the potential energy v of interaction between two molecules as a function of r; the quantity s (sigma) is the average diameter of the two molecules. Note that, when the intermolecular distance r is greater than 2 or 3 times the molecular diameter s, the intermolecular potential energy v is negligible. Intermolecular forces are gener-ally short-range. When r decreases below 3s, the potential energy decreases at first, indicating an attraction between the molecules, and then rapidly increases when r becomes close to s, indicating a strong repulsion. Molecules initially attract each other as they approach and then repel each other when they collide. The magnitude of intermolecular attractions increases as the size of the molecules increases, and it increases as the molecular dipole moments increase.
The average distance between centers of molecules in a gas at 1 atm and 25°C is about 35 Å (Prob. 2.55), where the angstrom (Å) is
⬅ 0.1 nm (2.87)*
Typical diameters of reasonably small molecules are 3 to 6 Å [see (15.26)]. The aver-age distance between gas molecules at 1 atm and 25°C is 6 to 12 times the molecular diameter. Since intermolecular forces are negligible for separations beyond 3 times the molecular diameter, the intermolecular forces in a gas at 1 atm and 25°C are quite small and make very little contribution to the internal energy U. Of course, the spatial distribution of gas molecules is not actually uniform, and even at 1 atm significant numbers of molecules are quite close together, so intermolecular forces contribute slightly to U. At 40 atm and 25°C, the average distance between gas molecules is only 10 Å, and intermolecular forces contribute substantially to U.
Let Uintermol,mbe the contribution of intermolecular interactions to Um. Uintermol,m differs for different gases, depending on the strength of the intermolecular forces.
Problem 4.22 shows that, for a gas, Uintermol,mis typically 1 to 10 cal/mol at 1 atm and 25°C, and 40 to 400 cal/mol at 40 atm and 25°C. (Uintermolis negative because intermolecular attractions lower the internal energy.) These numbers may be com-pared with the 25°C value Utr,m RT 900 cal/mol.
The fact that it is very hard to compress liquids and solids tells us that in con-densed phases the molecules are quite close to one another, with the average distance
3 2
1 Å⬅ 108 cm⬅ 1010 m
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between molecular centers being only slightly greater than the molecular diameter.
Here, intermolecular forces contribute very substantially to U. In a liquid, the molec-ular translational, rotational, and vibrational energies are, to a good approximation (Sec. 21.11), the same as in a gas at the same temperature. We can therefore find Uintermolin a liquid by measuring U when the liquid vaporizes to a low-pressure gas.
For common liquids, Umfor vaporization typically lies in the range 3 to 15 kcal/mol, indicating Uintermol,mvalues of 3000 to 15000 cal/mol, far greater in magnitude than Uintermol,min gases and Utr,min room-temperature liquids and gases.
Discussion of U in solids is complicated by the fact that there are several kinds of solids (see Sec. 23.3). Here, we consider only molecular solids, those in which the structural units are individual molecules, these molecules being held together by in-termolecular forces. In solids, the molecules generally don’t undergo translation or rotation, and the translational and rotational energies found in gases and liquids are absent. Vibrations within the individual molecules contribute to the internal energy. In addition, there is the contribution Uintermolof intermolecular interactions to the internal energy. Intermolecular interactions produce a potential-energy well (similar to that in Fig. 21.21a) within which each entire molecule as a unit undergoes a vibrationlike motion that involves both kinetic and potential energies. Estimates of Uintermol,mfrom heats of sublimation of solids to vapors indicate that for molecular crystals, Uintermol,m is in the same range as for liquids.
For a gas or liquid, the molar internal energy is
where Urest,mis the molar rest-mass energy of the electrons and nuclei, and is a con-stant. Provided no chemical reactions occur and the temperature is not extremely high, Uel,mis a constant. Uintermol,mis a function of T and P. Utr,m, Urot,m, and Uvib,mare func-tions of T.
For a perfect gas, Uintermol,m 0. The use of Utr,m RT, Urot,nonlin,m RT, and Urot,lin,m RT gives
(2.88) For monatomic gases (for example, He, Ne, Ar), Urot,m 0 Uvib,m, so
(2.89) The use of CV,m (Um/T)Vand CP,m CV,m R gives
(2.90) provided T is not extremely high.
For polyatomic gases, the translational contribution to CV,m is CV,tr,m R; the rotational contribution is CV,rot,lin,m R, CV,rot,nonlin,m R (provided T is not extremely low); CV,vib,mis a complicated function of T—for light diatomic molecules, CV,vib,mis negligible at room temperature.
Figure 2.15 plots CP,mat 1 atm versus T for several substances. Note that CP,m R 5 cal/(mol K) for He gas between 50 and 1000 K. For H2O gas, CP,mstarts at 4R 8 cal/(mol K) at 373 K and increases as T increases. CP,m 4R means CV,m 3R. The value 3R for this nonlinear molecule comes from CV,tr,m CV,rot,m R R. The increase above 3R as T increases is due to the contribution from CV,vib,m as excited vibrational levels become populated.
The high value of CP,mof liquid water compared with that for water vapor results from the contribution of intermolecular interactions to U. Usually CP for a liquid is substantially greater than that for the corresponding vapor.
The theory of heat capacities of solids will be discussed in Sec. 23.12. For all solids, CP,mgoes to zero as T goes to zero.
3
2 3
2
52
3
2
3
2
CV,m32R, CP,m52R perf. monatomic gas Um32RT const. perf. monatomic gas
Um32RT32RT 1or RT2 Uvib,m1T2 const. perf. gas
3 2 3
2
Um Utr,m Urot,m Uvib,m Uel,m Uintermol,m Urest,m
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The heat capacities CP,m (Hm/T)Pand CV,m (Um/T)Vare measures of how much energy must be added to a substance to produce a given temperature increase.
The more ways (translation, rotation, vibration, intermolecular interactions) a sub-stance has of absorbing added energy, the greater will be its CP,mand CV,mvalues.